General methods for computing (in theory, at least) the differential Galois group have been proposed by several authors [1]}, [2]}, [3]}, [4]} in the last past decades. However, those methods have an extremely high computational complexity and are therefore not practical for nontrivial problems. Moreover it is not clear how the differential Galois group of a given system depends on the coefficients of the latter. More recently an algorithm for computing the Lie algebra of the differential Galois group has been proposed in [5]}. This algorithm has a rather reasonable complexity, compared to the previous methods for computing the differential Galois group, but like the former it does not allow to clarify the relationship between the coefficients of the input system and its differential Lie algebra.
